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Lecture 09-2007 - Moments of More than One Random Variable...

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Moments of More than One Random Variable Lecture IX I. Covariance and Correlation A. Definition 4.3.1: , Cov X Y E X E X Y E Y E XY XE Y E X Y E X E Y E XY E X E Y E X E Y E X E Y E XY E X E Y 1. Note that this is simply a generalization of the standard variance formulation. Specifically, letting Y X yields: 2 2 Cov XX E XX E X E X E X E X 2. From a sample perspective, we have: 2 1 1 1 1 , n t t n t t t V X x n Cov X Y x y n 3. Together the variance and covariance matrices are typically written as a variance matrix: , , xx xy yx yy V X Cov X Y Cov X Y V Y Note that , , xy yx Cov X Y Cov Y X . 4. Substituting the sample measures into the variance matrix yields
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AEB 6933 Mathematical Statistics for Food and Resource Economics Lecture IX Professor Charles B. Moss Fall 2007 2 n t t t n t t t n t t t n t t t n t t t n t t t n t t t n t t t yy yx xy xx y y x y y x x x n y y n x y n y x n x x n s s s s S 1 1 1 1 1 1 1 1 1 1 1 1 1 The sample covariance matrix can then be written as: n n n n y x y x y y x x n 1 1 1 1 1 5. In terms of the theoretical distribution, the variance matrix can be written as: 2 2 , , , , x f x y dxdy xy f x y dxdy xy f x y dxdy y f x y dxdy 6. Example 4.3.2 X Y -1 0 1 1 0.167 0.083 0.167 0.417 0 0.083 0.000 0.083 0.167 -1 0.167 0.083 0.167 0.417 0.417 0.167 0.417
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